Thermoeconomic analysis of vapor power systems

NPS-59Nn75062A

tjitytir

NAVAL POSTGRADUATE SCHOOL

Monterey, California

THERMO ECONOMIC ANALYSIS OF VAPOR POWER SYSTEMS

by

CDR F. L. Sheppard, Jr., USN
J. K. Hartman
M. D. Kelleher
R. H. Nunn

30 June 1975

Approved for public release; distribution unlimited.

FEDDOCS
D 208.14/2:
NPS-59NN75062A

NAVAL POSTGRADUATE SCHOOL
Monterey, California

Rear Admiral Isham Linder J. R. Borsting

Superintendent Provost

THERMOECONOMIC ANALYSIS OF
VAPOR POWER SYSTEMS

A method is presented for determining the
relationships between the costs and technical per-
formance of vapor power systems in a manner which
permits fundamental design specifications to be
made optimally with respect to overall system life-
time costs. Means of applying optimization tech-
niques for large scale systems to the thermoeconomic
analysis of vapor power systems are described and
demonstrated with a simplified sample model. The
example studied is an environmentally driven ocean
thermal gradient system. A sequential unconstrain-
ed minimization algorithm is employed for overall
system design optimization.

The work reported herein has been supported by the
Energy Programs Office, Code L80, of the Civil
Engineering Laboratory, Port Hueneme, California;
work request N68305 75 WR-S-0068.

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■4. TITLE (end Subtitle)

THERM0EC0N0MIC ANALYSIS OF VAPOR POWER SYSTEMS

5. TYPE OF REPORT & PERIOD COVERED

FINAL, FY75

6. PERFORMING ORG. REPORT NUMBER

7. AUTHOR (a)

F. L. Sheppard, J. K. Hartman, M. D. Kelleher,
R. H. Nunn

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Naval Postgraduate School
Monterey, CA 93940

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N68305 75 WR-5-0068

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Civil Engineering Laboratory
Naval Construction Battalion Center
Port Hueneme, CA 93043

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30 Jun 75

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106

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19. KEY WORDS (Continue on reveree side if neceeeary end identify by block number)

Engineering Economics, Thermoeconomics , Thermodynamics, Optimization.

20. ABSTRACT (Continue on reveree eide if neceeeary and identify by biock number)

A method is presented for determining the relationships between
the costs and technical performance of vapor power systems in a
manner which permits fundamental design specifications to be
made optimally with respect to overall system lifetime costs.

Means of applying optimization techniques for large scale systems
to the thermoeconomic analysis of vapor power systems are described

1473 EDITION OF 1 NOV 65 IS OBSOLETE

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UU 1 JAN 73

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20. (continued) and demonstrated with a simplified sample model.

The example studied is an environmentally driven ocean thermal gradient
system. A sequential unconstrained minimization algorithm is employed
for overall systen design optimization.

DD Form 1473 (BACK)
. 1 Jan 73

S/N 0102-014-6601

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TABLE OF CONTENTS

I. INTRODUCTION 12

A. BACKGROUND 12

B. THERMOECONOMICS 17

C. OBJECTIVE 19

II. THERMOECONOMIC ANALYSIS 21

A. CONCEPT 21

B. PROBLEM REDUCTION 21

C. PROBLEM COORDINATION 2 3

III. SAMPLE ANALYSIS 2 9

A. PRELIMINARIES 2 9

B. BASIC SYSTEM DESCRIPTION 2 9

C. DETAILED DESCRIPTION OF ZONE 1 34

D. SOLUTION ALGORITHM 4 2

E. INITIAL COMPUTATIONAL CONSIDERATIONS 4 5

F. FIRST ZONE 1 SOLUTION 46

G. TESTING THE SOLUTION 50

H. ACCELERATION PROCEDURES 5 3

I. LINKING VARIABLE BEHAVIOR 54

J. ZONE VARIABLE BEHAVIOR 5 7

K. ZONE 1 SUMMARY 6 6

L. ZONE 2 ANALYSIS 6 6

M. COMPLETION OF THE ANALYSIS 68

IV. SUMMARY AND CONCLUSIONS 7 0

APPENDIX A - MODEL DEVELOPMENT 74

5

APPENDIX B - EXTENSION TO REAL SYSTEM

79

APPENDIX C - DATA 83

COMPUTER PROGRAM 98

REFERENCES 101

INITIAL DISTRIBUTION LIST 104

6

LIST OF TABLES

Table

III-l.

Table

III-2 .

Table

III-3 .

Table

III-4 .

Table

Ill- 5 .

Table

III-6 .

Table

III-7 .

Table

III-8 .

Table

III-9 .

Table

III-10

Table

IV-1.

Fluid Properties

Selected Values of Parameters

First Zone 1 Solution

Initial X Vectors
o

Fouling Factor Cost Effects
Dimensional Constraint Cost Effects
Cost Sensitivity to Modeling Equations
Improved Parameter Selections
Improved Zone 1 Solution
Results With Alternate Working Fluids
Summary of Design Evolution

Appendix

Table 1
Table 2
Table 3
Table 4

Appendix C
Appendix C
Appendix C
Appendix C

Results of T^ Investigation
Results of d Investigation
Results of Rp Investigation
Results of Dimensional Constraint
Investigation

7

LIST OF FIGURES

1. Vapor cycle diagram 12

2. Closed Rankine Cycle 14

3. Waste heat cycles 16

4. Model Coordination 25

5. Goal Coordination 2 7

6. Basic System 30

7. Heat Exchanger 32

8. T„ vs Cost, T = 5 5°F 56

9. T u vs Cost, T = 5 0°F , 55°F 58

n C

10. Cost vs Tube Diameter 60

11. Dimensional Constraint Effects 63

8

LIST OF SYMBOLS

Material Properties Units

p

working fluid density

lbm/ft 2

P H

seawater density

lbm/ft 2

C

D

working fluid heat capacity

Btu/lbm-°F

r

C u
pH

seawater heat capacity

Btu/lbm-°F

y

working fluid viscosity

lbm/ft-hr

y H

seawater viscosity

lbm/ft-hr

k

working fluid conductivity

Btu/hr-ft-°F

k H

seawater conductivity

Btu/hr-ft-°F

k

w

tube wall conductivity

Btu/hr-f t-°F

P

r

working fluid Prandtl number

dimensionless

P rH

seawater Prandtl number

dimensionless

Dimensions

d

tube inside diameter

ft

t

tube wall thickness

ft

S T

tube bank transverse spacing

ft

S L

tube bank longitudinal spacing

ft

l

heat exchanger length

ft

w

heat exchanger width

ft

a

heat exchanger height

ft

A

o

tube outside area

ft 2

A.

1

tube inside area

ft 2

A

mw

tube wall median area

ft 2

9

Heat Exchanger Characteristics

Units

area

U u overall heat transfer coefficient
n

number of thermal units

intermediate performance variable
n

9^ rate/capacity ratio

h outside heat transfer coefficient

0

h. inside heat transfer coefficient

1

Rp fouling resistance

R tube wall resistance

w

R. inside resistance

1

R outside resistance

o

N' number of tube rows

N T total number of tubes

N u working fluid Nusselt number

N u seawater Nusselt number
uH

Temperatures

Tp working fluid: exchanger outlet

T c working fluid: exchanger inlet

Tr£ hot seawater
T n _ cold seawater

Power

G gross power output

E net power output

Wp working fluid pumping power
Wp seawater pumping power

Flow parameters

m working fluid flow rate

nip seawater flow rate

G^ working fluid mass flux

seawater velocity

Btu/ft 2 -hr-°F
dimensionless
dimensionless
dimensionless
Btu/ft 2 -hr-°F
Btu/ft 2 -hr-°F
ft 2 -hr-°F/Btu
ft 2 -hr-°F/Btu
ft 2 -hr-°F/Btu
ft 2 -hr-°F/Btu
dimensionless
dimensionless
dimensionless
dimensionless

°F

op

°F

°F

MW

MW

KW

KW

lbm/hr

lbm/hr

lbm/hr-ft 2

ft/hr

10

Units

Flow parameters (cont'd)

R e ^ seawater Reynolds number
R g c working fluid Reynolds number
DP friction head, working fluid

DPj| friction head, seawater

f friction factor, working fluid

f u friction factor, seawater

n

Costing Factors
it profit

z^ capital cost, heat exchanger
capital cost
z^2 capital cost
Z£ capital cost, all pumps
COST total zone 1 cost
p Q market price of power
C^ capacity-head product
C^ capacity-head product

base cost seawater pump
B£2 base cost working fluid pump
P^ intermediate cost variable
P^2 intermediate cost variable
F^ material factor
F^2 material factor
D ^ price index

Miscellaneous

^ energy conversion factor
p working fluid pump efficiency
n H seawater pump efficiency
X vector of decision variables

dimensionless

dimensionless

lbf/ft 2

lbf/ft 2

dimensionless

dimensionless

$

$

$

$

$

$

$-hr/ f t-lbf
gal-lbf /min-in 2
gal-lbf /min-in 2
$

$

$

$

dimensionless

dimensionless

dimensionless

MW-hr/lbm-°F

dimensionless

dimensionless

various

11

I. INTRODUCTION

A. BACKGROUND

The motivation for developing new energy technology, long
forecast by such researchers as Putnam [1], has now become so
widely understood as to require no elaboration here. Research
and development efforts are proceeding on a broad front in
search of alternatives to the conventional non-renewable fossil
fuels and potentially hazardous fission processes.

One class of proposals seeks to extract useful energy
from sources existing in nature. With the exception of geo-
thermal energy, virtually all of these rely ultimately on some
phenomenon associated with receipt by the earth of energy
radiated from the sun. A sub-class of these "environmental"
energy systems utilize vapor cycles in mechanizing the
conversion from the diffuse heat sources found in nature into
the more concentrated and transportable energy forms required
in many applications.

12

Figure 1 diagrams the fundamental concept upon which these
environmental vapor power systems depend. Heat is extracted
from some natural source of elevated temperature (geothermal
wells, direct solar collectors, hot seawater, etc.) and
transferred to a working fluid. Devices suitable to the
application convert the thermodynamically available portion of
the available heat into other forms of energy, and the
unavailable energy is rejected to a thermal sink. Both open
and closed vapor cycles are possible [2], [3] and the products
of conversion can take many forms , including electricity and
fuels such as hydrogen, methanol, and ammonia [4].

One widely used vapor cycle is the closed Rankine cycle

[5] , shown schematically in Figure 2.

In addition to the conservation of chemical fuels,
certain of the proposed methods of harnessing energy hold
promise of significant additional advantages. It is expected
that their effects on the environment will be relatively benign

[6] , particularly in terms of atmospheric and thermal pollution.
They would cause no addition to the total heat burden at the
earth's atmosphere and rely on sources which are continuously
renewed by natural processes [7].

There is another class of vapor power cycles which,
although not always exploiting environmental energy sources,
shares enough of the operating characteristics of those that
do to warrant mention as a group amenable to the type of analy-
sis discussed in this paper. These are the "bottoming" cycles
for extracting power from the still energetic discharges of

13

FIGURE 2 . Closed Rankine Cycle

geothermal, nuclear, and chemically fueled plants. Figure 3
shows two general types of these "waste heat"cycles.

The essential feature which differentiates both the
environmental and waste heat vapor systems from conventional
ones is the relatively low thermal potential within which
they operate. A consequence of this characteristic is that
the size scale of all the cycle components is increased in
comparison with conventional plants. Heat exchange surface
areas must be enlarged for sufficient heat to be transferred
through small driving potentials , and with less energy avail-
able from each unit of fluid circulated, a far greater volume
rate of working fluid must be cycled. Pumps, pipes, and
conversion devices such as turbines all grow in size as the
temperature difference between the source and sink is reduced
while the energy product is held constant.

Viewed fundamentally, environmental power systems employ
technology which has been available for many years. Many
concepts have been tested with working models or demonstration
plants, and some are employed presently on a small scale.
Although significant technical problems arise in connection
with specific applications, these do not appear to be permanen
obstacles. Net energy asessments appear favorable and
questions of material availability and local adverse environ-
mental effects seem amenable to solution [6].

The primary question which will determine when environ-
mental energy sources will be exploited on a scale large
enough to significantly affect the energy market is that of

15

FIGURE 3. Waste Heat Cycles

16

system economics. Although some researchers predict plant
costs which are currently competitive with conventional
methods, [9] uncertainties arising from the lack of operating
experience weaken the claims of these proponents. Until
economic viability can be conclusively shown, risk aversity
will act as a strong deterrent to attracting the very large
amounts of venture capital required. With the private sector
presently unconvinced, the federal government is undertaking
the expenditures required for research and development efforts
[ 10 ] .

B. THERMOECONOMICS

Typically, system design and cost studies are conducted
in a two-step or, at best, iterative process. Designers
assemble specifications based on technically achievable and
desirable functional characteristics. They are, of course,
guided in their design decisions by some measure of intuition
as to the economic impacts, usually based on prior experience
with similar programs. The degree of detail in the initial
specifications presented, in fact, often reflects the confi-
dence held by the engineers in their economic appraisals. The
system and component specifications are then subjected to cost
analysis, prime cost factors are identified, and technical-
economic trade-offs are suggested.

The problems arising from this partial separation of the
design and costing steps are more or less severe according to
the application. There is a fundamental difficulty in
communicating the two groups' understandings in a meaningful

17

way, a difficulty which increases as the novelty of the design
situation and hence number of unconstrained design choices
increases. In their quest for a sophisticated design product,
engineers may design around some apparently desirable
parameter, such as a high heat transfer coefficient. Cost
analysts may take this figure as fixed and address themselves
to questions of material selection, maintainability, or
manufacturing tolerances without recognizing that adjusting
the heat transfer coefficient itself could produce the most
rewarding cost effects.

This type of difficulty is most severe when little ex-
perience is available to guide the engineer's fundamental
choices, as is true in the case of environmental vapor power
systems. The small available thermal potential drastically
distinguishes these systems from their high temperature
counterparts. In a fuel-fired generating plant, for example,
the power required to pump the working fluid can be neglected
in a first approximation, and a variation of 1°F temperature
difference across a boiler tube is insignificant. As is
demonstrated later in this study, such considerations can have
a profound influence on the overall economics of an environ-
mental plant.

To some extent, the engineer's problem can be viewed as
being where to start. Recognizing that pumping power is going
to be substantial, he might decide to assign 10 or 20 percent
of the plant's overall output to pumping requirements and
build much of the rest of the design about this choice. Or

18

he might choose to utilize 30 percent of the total temperature
difference for heat transfer, leaving the remaining 70 percent
available for enthalpy drop across the turbine. He might
establish a dimensional constraint, based on nothing much more
than the feeling that a 100 foot diameter pipe is a very big
pipe .

Unf ortunatly , all these basic choices involve performance
and cost tradeoffs. If flow rates are increased to enhance
heat transfer, drag coefficients increase as well. How much
improvement in heat transfer is worth how large an increase in
pump head, and hence pump work? Pump work is also influenced
by heat exchanger tube diameter, spacing, and surface charac-
teristics, which also affect space and material requirements.
How much should one be willing to pay to reduce fouling heat
resistance? If heat exchange is dominated by fouling
resistance, is it worth the extra temperature drop necessary
to shift to a different boiling regime? Unless the cost
analyst is knowledgeable about the thermodynamic consequences
of costing factors he is in as poor a position as the engineer
to make the tradeoffs in dollars per millimeter of fouling
organisms .

C. OBJECTIVE

What is needed is an analytical method whereby overall
economic effects may be integrated into engineering design in
such a way that the designer's intuition may be enhanced in
trading off the costs and benefits of parameter selection at
the margin. A means is required for mapping the large number

19

of interrelated engineering variables into their individual
and collective effects in the marketplace, where the ultimate
design appraisal will take place.

The research reported on in this paper is intended to
develop and evaluate one method of integrating marginal
cost/benefit analysis into engineering design and to show
the kinds of information which could thus be gained. In this
initial investigation, no effort has been made to apply the
method to any particular practical design problem or to
produce analytical insights into existing systems. The
intent has been to show how thermoeconomic analysis can be
performed and what value it can have when applied to a specific
real case.

20

II. THERMOECONOMIC ANALYSTS

A. CONCEPT

Profit is the difference between benefits and costs,
both broadly considered. When these can be related over a
common set of decision variables, X, one may write

tt (X) = B(X) - C(X)

with B(X) representing the sum of all benefits, and C(X) the
sum of all costs:

N

B (X) = $ B . (X)

1 1

N

C(X) = 2 C.(X)

• 1
i

If all the relevant B^ and C^ can be defined functionally
over X, performing

maximize: ir

subject to: a required level of performance (A)

would produce the desired optimization.

B. PROBLEM REDUCTION

Attempting a global optimization directly with all possible
costs and benefits considered, although theoretically possible,
encounters many practical difficulties [11]. It is possible,
however, to achieve considerable reduction of the problem
without sacrificing many of the benefits of the analysis.

21

First, although the impacts of general (and difficult to
quantify) externalities, such as independence from foreign
control of energy sources, are important and should not be
excluded from the final analysis, many interior decisions
suffer not at all from excluding externalities such as these
from most of the study. Many other factors are not related
to the decision variables (X^) and therefore do not affect
marginal design choices. For example, personnel training
expenses are not close functions of tube diameter. Since
the solution to

maximize it = B(X) - C(X) - D
subject to g(X) = 0

where D is constant with respect to X is identical to the
solution of

maximize tt = B(X) - C(X)
subject to gCX) = 0

any factor which acts only as an additive constant may be
excluded from the analysis without affecting the results.

Even with invariants over the decision variables ex-
cluded, there are other serious impediments to seeking global
solutions to (A) . Convexity of the optimization problem is
not assured by the physical relations modeled.^" When all

A convex optimization problem is defined as one with a
convex objective function, to be minimized , concave £ inequal-
ity constraints, and linear equality constraints [12]. The
conditions on the constraints assure that the feasible region
is a convex set, i.e., for every X, 0 < X < 1, and any two
points X-p X£ €T, a convex set, [XX^ + (l-X^^] €T.

22

decision variables are considered at once in a global assault
it is increasingly difficult to test for uniqueness of the
solution. Secondly, since design variables in one system
component often are only distantly related to those in another,
insights are obscured when they are varied simultaneously
within one code. Thirdly, the model can never be exact. It
is important for the designer to keep track of the effects
of his. modeling choices in detail. This is more easily achieved
by putting the pieces together sequentially than all at once.
Finally, the designer often has adequate information available
to intelligently fix some of the variables. It is unnecessary
to complicate the analysis by including as free variables
factors which are closely constrained by other considerations.

For these reasons, it appeared desirable to follow the
usual procedure for the optimization of large scale systems
by decomposing the problem into coherent interrelated zones
and achieving global optimality through one of the available
zone coordination methods. The next section contains an
outline of the general procedure.

C. PROBLEM COORDINATION

The general theory for optimizing large scale systems
through coordination of smaller subsystems can be found in
references such as Wismer [11], The following discussion of
the two basic approaches is greatly particularized in that
the terminology and composition of the examples reflect the
structure of the sample analysis which is presented in section
III.

23

The first approach, called the model coordination method,
can be understood through consideration of the decomposed
system shown in Figure 4.

Define y = (T^,T^) as a vector of coordinating variables
and X-^ and X^ as vectors of design variables in zones 1 and 2.
Then construct the zone subproblems:

minimize: f^(X^,y)

X i5 y i = 1,2

subject to: g^(X^,y) >_ 0
h. (X. ,y ) = 0.

l l

The first level of analysis is conducted by setting y =
y°, a feasible value of y. Then solve

minimize: f^CX^y 0 )

X. i = 1,2

l ’

subject to: g^(X^,y°) >_ 0
h i (X i ,y°) = 0.

The solutions are designated X^ and X^. The second level
of analysis seeks to find the value of y which produces the
minimum value of

F(X^, x\ 9 y) = f^xj, y) + f^x], y)

subject to: g^(X^,y) >_ 0 i = 1,2

h i (X^,y) = 0.

Designate the solution by y^. An iterative sequence is

now established by replacing y° by y"*" in the first level

. . ~2 ~2
problems and resolving; using the resulting X^ to find y

24

T

H

FIGURE 4. Model Coordination

25

in the second level problem, and so forth until the improve-
ment achieved with each iteration is less than a specified
tolerance .

This approach is called the model coordination method
because the task of the second level control is to choose the
linking variables in such a way that the independent first
level systems are forced to choose solutions which in fact
correspond to an overall system optimum. In some references,
this is called the feasible method.

The second method, called goal coordination or the dual
feasible method, views the decomposed system as in Figure
5 .

It is important to note that in this formulation of the
problem, y does not necessarily equal z. The interactions
have been literally removed by "cutting" all links between
subsystems .

The physical requirement that, in the end, y must equal
z, termed the interaction-balance principle, is satisfied in
the course of the analysis as follows.

In the first level analysis, let X =X°. Then solve

minimize :

L 1

«1

> t h

•r c -.i°)

= f l

subject to:

Si

(X 1

> t h

>*'c ) i

0

h .
1

(X 1

,t h

,5T C > =

0

minimize :

L 2

(x 2

»*H

= f 2

<X 2

)-X°X- H ♦ A°

subject to:

§2

(X 2

,5T h

H

0

1 V

0

h 2

(X 2

,h h

’V -

0

26

T

l E

~ 1Xli *T

f

ZONE 1

ZONE 2

X 1

x 2

~1

h {

X" c

i 2j ~

C

FIGURE 5 . Goal Coordination

Define :

l - ! T H> T ci

z = jjr H .x- e j

X = S x i> x jj

27

yielding: X^ , , and y^ .

The second level problem then becomes choosing A such
that solutions to the first level problems result in
satisfaction of the interaction balance principle. This is a
well behaved optimization in its own right and is solved with
the usual techniques of mathematical programming.

Notice that in this method the coordination effect of
the second level analysis is effected by manipulating the
goals of the first level analysis through adjustment of the A
coordinating variables, hence the term goal coordination
method. The A multipliers enter the individual first-level
problem objective functions linearly and act like prices,
adding to or subtracting from the performance function of each
subproblem in direct proportion (with proper sign) to the amount
of demanded and the amount of y^ produced. Thus the
second-level goal coordination can be interpreted as modifying
"prices" of the interacting variables in order to force the
independent first-level problems to select consistent values
of the linking variables and hence the correct overall system
optimum .

Much additional information is available in the results
of the steps of the solution when cast in this format, and the
interested reader is referred to the considerable literature
on the subject, [13, 14, 15 and 16, for instance].

Because of its more straight forward formulation, the
sample analysis in the following section is cast in the model
coordination format.

28

III. SAMPLE ANALYSIS

A. PRELIMINARIES

The methodology of thermoeconomic analysis can best be
described through demonstration with a sample analysis. For
this purpose, an extremely simplified thermal system was
selected; one which contains the essential features of a
realistic system but avoids a number of complications which
would tend to obscure the technique. It should be well under-
stood that with lumped component representations and several
significant losses neglected, the model chosen can not be
treated as representing a practical plant, nor can the
results of the analysis be taken as having implications for a
real system. The model does have many similarities with ocean
thermal energy conversion plants as presently conceived, and
in Appendix B a discussion is presented as to what refine-
ments would be necessary to extend the sample model into one
of a functional ocean thermal system. For ease of exposition,
the model will be discussed without repeated references to
these departures from realism.

B. BASIC SYSTEM DESCRIPTION

Figure 6 diagrams the basic system considered. The
thermal source consists of an infinite supply of seawater at
a temperature of T^ = 85°F. The thermal sink is a similarly
limitless supply of seawater at T^ = 45 °F. In the energy
extraction component the ammonia working fluid is heated as

29

FIGURE 6 . Basic System

it flows through the shell side of a rectangular crossflow
shell and tube heat exchanger with smooth staggered tubes.
The heat is provided by relatively hot seawater flowing
through the tubes as shown in Figure 7.

Single phase heat exchange takes place in the device ,
with both fluids remaining compressed liquids.

The energy conversion component receives hot ammonia

liquid from the heater, accomplishes energy conversion to

electrical form, and discharges the liquid at a lower

temperature. The manner in which the conversion takes

place is unspecified and not necessary for this sample

analysis. The conversion process is described by a single

parameter, ip , which measures how much energy is converted

to electricity per pound mass of ammonia flowing through the

device per degree Fahrenheit temperature drop. The value

selected for was 0.5 Btu/lbm °F, which is approximately

half the specific heat for liquid ammonia and, incidentally

about the same energy available to a turbine with saturated

. . 2

vapor inlet conditions.

The fluid pressurizer consists simply of one or more
standard centrifugal pumps, sufficient to drive the working
fluid through the system at the required rate. Both the hot
and cold seawater are similarily pumped.

2

Saturated ammonia vapor at 80 °F has enthalpy of 630
Btu/lbm. Isentropic expansion to 50°F results in enthalpy
of 615 Btu/lbm, or 0.5 Btu/lbm per °F [17].

31

AMMONIA

Ji

PZ

w

Eh
Eh <
O ^
K <
W
cn

32

FIGURE 7 . Heat Exchanger

The system was zoned as depicted in Figure 6, with
zone 1 consisting of the heat exchanger and working fluid and
hot seawater pumps , and zone 2 consisting of the energy
conversion device and the cold seawater pumps.

Before proceeding further, it should be made clear that
none of the above assumptions nor those which follow consti-
tute final arbitrary design selections. Each parameter,
fluid, and configuration is eventually fixed as an output of
the analysis itself. Their initial specification should be
regarded as tentative, pending further information to be
developed in the course of the study. This preliminary
configuration acts only as a starting point.

The next step is to characterize zonal inputs and outputs
in terms of appropriate physical variables which are descrip-
tive of the transactions taking place at zone boundaries .

The principal feature of the hot seawater is its temperature,
Tr£5 so this was chosen as the input to zone 1 from the thermal
source. The other input to zone 1 is the ammonia discharge
from zone 2, which is again described by its temperature, T .
The output of zone 1 is hot ammonia liquid at temperature T^.
The only remaining variables which cross zone boundaries are
the electrical output of zone 2, G, and the cold seawater
from the thermal sink at temperature T^.

The global problem is to maximize the profit obtainable
by selling the system’s electrical output at market prices.
Translated into zone terms , this implies that each zone should
produce the required level of output at minimum cost, given
the inputs it has to work with.

33

C. DETAILED DESCRIPTION OF ZONE 1

The heat exchanger has length in the direction of sea-
water flow (£), height in the direction of working fluid flow
(a), and width transverse to each (w) . Tubes have inside
diameter (d), wall thickness t, and have transverse and
longitudinal spacing S^, and S^. The inside and outside heat
flow resistances due to chemical and biological fouling are
combined into one fouling resistance, Rp.

Selection of ammonia as the working fluid and seawater
as the heat source leads to the following table of approx-
imate physical properties, all considered constant.

Table III-l.
FLUID PROPERTIES

SPECIFIC

FLUID

DENSITY

(lbm/ft3)

VISCOCITY
( lbm/ft-hr )

CONDUCTIVITY

(Btu/hr-ft-°F)

HEAT

(Btu/lbm°F)

Ammonia

p = 40

V = 0.5616

K = 0.307

C = 1.135
P

Seawater

J-

CD

II

m

Cl

V - 2.37

K h = 0.349

C =1.0
P

The working fluid mass flowrate (m) and hot seawater mass
flowrate (m^) are provided by centrifugal pumps, which deliver
the required flows against the head created by frictional and
form losses in the heat exchanger, (minor losses were
neglected but could easily be included) . The pumping power
for these pumps, and , is a parasitic deduction from the
gross plant electrical output, G^.

34

Fixing the input, output, and linking variables (T HE ,

^CE’ ^H’ T c’ an< ^ ^ temporarily helps to focus an understand-
ing of zone 1 objectives and constraints. T^ E and T^ E were
set previously at 85°F and 45°F, and these define the thermal
potential available to the system. If half of this potential
is assigned to drive heat through the exchanger surfaces,
and T c selections of 75°F and 55°F result. If overall power
output is set at 25 mw (a frequently encountered figure for
prototype ocean thermal plants), the required working fluid
flow rate can now be determined from the relationship

G = m iKT u -T ) • (1)

He

It now becomes evident that the task of zone 1 is to
receive T^ E and T c and produce the required m at with
minimum cost. The next major task is to select the design
variables to be used. To do this, we first look at the
governing physical and cost relationships.

As discussed in [18] the performance of a heat exchanger
can be described in terms of its effectiveness:

€ =

T u - T
H c

T - T
HE c

= 1 - e

-re

where

r = 1 - e

-NO

and

N =

¥h

mC

0 = ” hC p h

mC

Alternatively, the amount of heat transferred can be found from

35

Q = U H A H AT LM-

In either case, the fundamental process description is in
terms of heat exchange surface area, heat exchange coeffic-
ients, flowrates, temperatures, and fluid properties.

The major costs in zone 1 are the capital costs of the
heat exchanger, z^, the pumps, z^, and the cost of the
pumping power. This latter can be considered as an oppor-
tunity cost and valued at the amount which could have been
realized had the parasitic pumping power, and W^, been
sold at the prevailing rate, p , instead of being used
internally. Correlations are available [20] which give
capital costs of heat exchangers as functions of heat
exchanger area and capital costs of pumps in terms of the
product of flowrate and head. The frictional head is usually
determined empirically and related to flow velocities,

exchanger configuration, and fluid properties in terms of

3

Reynolds numbers.

It might initially appear attractive to choose the
design variables to be m^, U^, , and the friction heads

3

The Reynolds number is a dimensionless grouping of
physical variables which indicates the ratio of inertia
forces to viscous forces. It is formed from the product
of a characteristic velocity times a characteristic
dimension, divided by the fluid kinematic viscocity [21],

The Prandtl number is also dimensionless, being a
measure of the ratio of the diffusivity of momentum to
the diffusivity of heat. It is formed by multiplying
the fluid’s specific heat times its viscocity and
dividing by its conductivity [21].

36

DP and DP^ , but it is at this point that a key feature of
thermoeconomic analysis comes into force. Recall that what
is desired is a way to discover the tradeoffs between cost
and performance. Whatever variables are chosen, it must be
possible to establish this balance through the functional
relations over the domain of the variable set. As an
example of what happens otherwise, consider working with the
variables suggested just above. Performance can always be
improved by increasing U^, and it doesn't cost anything to
do so since is absent from the cost correlation. Costs
can always be decreased by dropping the DP's, and performance
would seem to be unaffected because the heat exchange equations
do not contain pressure drop terms. Any sensible computer
code would therefore drive and DP as high and low,
respectively, as is allowed. Setting a constraint on these
parameters is, in effect, arbitrarily choosing them, and no
information has been gained in the process. Achieving high
heat transfer with low pressure drops is known to be
desirable a priori .

Besides not permitting a cost-performance balance to be
weighed, there is a second problem with the variable list
suggested, namely that and the DP's are inextricably linked
through the Reynolds numbers. With a given working fluid,
heat transfer can only be improved by raising the Reynolds
numbers with the concurrent result that the pressure drops
are increased simultaneously. This is the core concern of
the zone 1 analysis: is made up of the inside coefficient,

37

h-, the outside coefficient, h , the tube wall conductivity
and the fouling resistance. There are two pressure drops of
concern, one in the seawater and one in the working fluid.

What combination of these parameters will produce the heat
exchange required at minimum cost?

To identify an appropriate variable set over which to
find this cost-performance balance, one must look to the
next level. Besides physical fluid properties, Reynolds
numbers depend upon flow rates and spatial dimensions.

Surface area depends on spatial dimensions. Pump work depends
on flow rate and pressure drop, which vary as functions of
Reynolds number.

Clearly, then, all the cost and performance calculations
can be built up in terms of flow rates and spatial dimensions ,
and this is the highest level of variable with which the
desired tradeoff can be made with the functional relationships
available. Note, however, that if other valid relationships
could be found which gave the information required in terms of
other quantities, other variables might be able to be used.

With the tools at hand, though, it was decided to perform
the analysis in terms of dimensions and flowrates. But
variable selection is not yet complete; one has the option of
which parameters will be allowed to vary independently in the
code and which will be controlled externally. This question
is resolved as a matter of judgment, and depends upon the
confidence the designer has in his preliminary intuition and
what specific information he seeks from the analysis. Hope-

38

fully, this matter will be clarified as the analysis proceeds.
For the present investigation, it was decided to permit four
variables to "float": seawater flow rate, and the length,

height and width of the heat exchanger. The following
parameter selections were made. Included are those which
have been discussed previously.

The functional relationships over both costs and
performance are developed in detail in Appendix A and summar-
ized below.

Pump Work, Working Fluid

Table III-2.

SELECTED VALUES OF PARAMETERS

working fluid:
tube diameter:
tube wall thickness:
tube spacing, S^:
tube spacing, S. :

ammonia

1/2 inch
0.035 inch
1.5 d inches
1.5 d inches
8 5°F
7 5 ° F
55 °F
2 5 MW

0.5 Btu/lbm-°F
0 . 0 3/Kw-hr
0.9

0.005 ft 2o F" hr /Btu
30 Btu/ft-hr-°F

hot seawater temperature:
hot working fluid temperature:
cold working fluid temperature:
Gross plant power output:
energy conversion factor,
market price of energy:
pump efficiencies:
fouling heat resistance:
tube wall conductivity:

WP = mD £

PR

39